src/HOL/Tools/res_axioms.ML
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(*  Author: Jia Meng, Cambridge University Computer Laboratory
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    ID: $Id$
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    Copyright 2004 University of Cambridge
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Transformation of axiom rules (elim/intro/etc) into CNF forms.    
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*)
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signature RES_AXIOMS =
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  sig
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  exception ELIMR2FOL of string
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  val tagging_enabled : bool
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  val elimRule_tac : thm -> Tactical.tactic
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  val elimR2Fol : thm -> term
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  val transform_elim : thm -> thm
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  val clausify_axiom_pairs : (string*thm) -> (ResClause.clause*thm) list
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  val cnf_axiom : (string * thm) -> thm list
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  val meta_cnf_axiom : thm -> thm list
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  val rm_Eps : (term * term) list -> thm list -> term list
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  val claset_rules_of_thy : theory -> (string * thm) list
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  val simpset_rules_of_thy : theory -> (string * thm) list
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  val claset_rules_of_ctxt: Proof.context -> (string * thm) list
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  val simpset_rules_of_ctxt : Proof.context -> (string * thm) list
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  val clausify_rules_pairs : 
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        (string*thm) list -> thm list -> (ResClause.clause*thm) list list * thm list
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  val clause_setup : (theory -> theory) list
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  val meson_method_setup : (theory -> theory) list
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  end;
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structure ResAxioms : RES_AXIOMS =
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struct
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val tagging_enabled = false (*compile_time option*)
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(**** Transformation of Elimination Rules into First-Order Formulas****)
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(* a tactic used to prove an elim-rule. *)
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fun elimRule_tac th =
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    ((rtac impI 1) ORELSE (rtac notI 1)) THEN (etac th 1) THEN
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    REPEAT(fast_tac HOL_cs 1);
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exception ELIMR2FOL of string;
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(* functions used to construct a formula *)
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fun make_disjs [x] = x
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  | make_disjs (x :: xs) = HOLogic.mk_disj(x, make_disjs xs)
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fun make_conjs [x] = x
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  | make_conjs (x :: xs) =  HOLogic.mk_conj(x, make_conjs xs)
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fun add_EX tm [] = tm
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  | add_EX tm ((x,xtp)::xs) = add_EX (HOLogic.exists_const xtp $ Abs(x,xtp,tm)) xs;
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fun is_neg (Const("Trueprop",_) $ (Const("Not",_) $ Free(p,_))) (Const("Trueprop",_) $ Free(q,_)) = (p = q)
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  | is_neg _ _ = false;
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exception STRIP_CONCL;
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fun strip_concl' prems bvs (Const ("==>",_) $ P $ Q) =
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      let val P' = HOLogic.dest_Trueprop P
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  	  val prems' = P'::prems
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      in
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	strip_concl' prems' bvs  Q
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      end
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  | strip_concl' prems bvs P = 
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      let val P' = HOLogic.Not $ (HOLogic.dest_Trueprop P)
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      in
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	add_EX (make_conjs (P'::prems)) bvs
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      end;
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fun strip_concl prems bvs concl (Const ("all", _) $ Abs (x,xtp,body))  = strip_concl prems ((x,xtp)::bvs) concl body
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  | strip_concl prems bvs concl (Const ("==>",_) $ P $ Q) =
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    if (is_neg P concl) then (strip_concl' prems bvs Q)
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    else
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	(let val P' = HOLogic.dest_Trueprop P
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	     val prems' = P'::prems
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	 in
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	     strip_concl prems' bvs  concl Q
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	 end)
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  | strip_concl prems bvs concl _ = add_EX (make_conjs prems) bvs;
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fun trans_elim (main,others,concl) =
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    let val others' = map (strip_concl [] [] concl) others
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	val disjs = make_disjs others'
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    in
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	HOLogic.mk_imp (HOLogic.dest_Trueprop main, disjs)
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    end;
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(* aux function of elim2Fol, take away predicate variable. *)
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fun elimR2Fol_aux prems concl = 
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    let val nprems = length prems
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	val main = hd prems
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    in
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	if (nprems = 1) then HOLogic.Not $ (HOLogic.dest_Trueprop main)
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        else trans_elim (main, tl prems, concl)
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    end;
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(* convert an elim rule into an equivalent formula, of type term. *)
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fun elimR2Fol elimR = 
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    let val elimR' = Drule.freeze_all elimR
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	val (prems,concl) = (prems_of elimR', concl_of elimR')
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    in
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	case concl of Const("Trueprop",_) $ Free(_,Type("bool",[])) 
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		      => HOLogic.mk_Trueprop (elimR2Fol_aux prems concl)
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                    | Free(x,Type("prop",[])) => HOLogic.mk_Trueprop(elimR2Fol_aux prems concl) 
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		    | _ => raise ELIMR2FOL("Not an elimination rule!")
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    end;
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(* check if a rule is an elim rule *)
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fun is_elimR th = 
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    case (concl_of th) of (Const ("Trueprop", _) $ Var (idx,_)) => true
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			 | Var(indx,Type("prop",[])) => true
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			 | _ => false;
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(* convert an elim-rule into an equivalent theorem that does not have the 
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   predicate variable.  Leave other theorems unchanged.*) 
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fun transform_elim th =
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  if is_elimR th then
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    let val tm = elimR2Fol th
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	val ctm = cterm_of (sign_of_thm th) tm	
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    in
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	prove_goalw_cterm [] ctm (fn prems => [elimRule_tac th])
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    end
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 else th;
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(**** Transformation of Clasets and Simpsets into First-Order Axioms ****)
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(* repeated resolution *)
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fun repeat_RS thm1 thm2 =
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    let val thm1' =  thm1 RS thm2 handle THM _ => thm1
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    in
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	if eq_thm(thm1,thm1') then thm1' else (repeat_RS thm1' thm2)
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    end;
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(*Convert a theorem into NNF and also skolemize it. Original version, using
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  Hilbert's epsilon in the resulting clauses.*)
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fun skolem_axiom th = 
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  let val th' = (skolemize o make_nnf o ObjectLogic.atomize_thm o Drule.freeze_all) th
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  in  repeat_RS th' someI_ex
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  end;
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fun cnf_rule th = make_clauses [skolem_axiom (transform_elim th)];
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(*Transfer a theorem into theory Reconstruction.thy if it is not already
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  inside that theory -- because it's needed for Skolemization *)
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(*This will refer to the final version of theory Reconstruction.*)
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val recon_thy_ref = Theory.self_ref (the_context ());  
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(*If called while Reconstruction is being created, it will transfer to the
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  current version. If called afterward, it will transfer to the final version.*)
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fun transfer_to_Reconstruction th =
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    transfer (Theory.deref recon_thy_ref) th handle THM _ => th;
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fun is_taut th =
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      case (prop_of th) of
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           (Const ("Trueprop", _) $ Const ("True", _)) => true
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         | _ => false;
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(* remove tautologous clauses *)
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val rm_redundant_cls = List.filter (not o is_taut);
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(* transform an Isabelle thm into CNF *)
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fun cnf_axiom_aux th =
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    map zero_var_indexes
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        (rm_redundant_cls (cnf_rule (transfer_to_Reconstruction th)));
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(**** SKOLEMIZATION BY INFERENCE (lcp) ****)
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(*Traverse a term, accumulating Skolem function definitions.*)
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fun declare_skofuns s t thy =
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  let fun dec_sko (Const ("Ex",_) $ (xtp as Abs(_,T,p))) (n, (thy, axs)) =
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	    (*Existential: declare a Skolem function, then insert into body and continue*)
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	    let val cname = s ^ "_" ^ Int.toString n
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		val args = term_frees xtp  (*get the formal parameter list*)
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		val Ts = map type_of args
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		val cT = Ts ---> T
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		val c = Const (Sign.full_name (Theory.sign_of thy) cname, cT)
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		val rhs = list_abs_free (map dest_Free args, HOLogic.choice_const T $ xtp)
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		        (*Forms a lambda-abstraction over the formal parameters*)
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		val def = equals cT $ c $ rhs
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		val thy' = Theory.add_consts_i [(cname, cT, NoSyn)] thy
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		           (*Theory is augmented with the constant, then its def*)
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		val cdef = cname ^ "_def"
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		val thy'' = Theory.add_defs_i false [(cdef, def)] thy'
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	    in dec_sko (subst_bound (list_comb(c,args), p)) 
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	               (n+1, (thy'', get_axiom thy'' cdef :: axs)) 
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	    end
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	| dec_sko (Const ("All",_) $ (xtp as Abs(a,T,p))) (n, thx) =
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	    (*Universal quant: insert a free variable into body and continue*)
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	    let val fname = variant (add_term_names (p,[])) a
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	    in dec_sko (subst_bound (Free(fname,T), p)) (n, thx) end
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	| dec_sko (Const ("op &", _) $ p $ q) nthy = 
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	    dec_sko q (dec_sko p nthy)
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	| dec_sko (Const ("op |", _) $ p $ q) nthy = 
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	    dec_sko q (dec_sko p nthy)
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	| dec_sko (Const ("HOL.tag", _) $ p) nthy = 
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	    dec_sko p nthy
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	| dec_sko (Const ("Trueprop", _) $ p) nthy = 
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	    dec_sko p nthy
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	| dec_sko t nthx = nthx (*Do nothing otherwise*)
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  in  #2 (dec_sko t (1, (thy,[])))  end;
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(*cterms are used throughout for efficiency*)
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val cTrueprop = Thm.cterm_of (Theory.sign_of HOL.thy) HOLogic.Trueprop;
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(*cterm version of mk_cTrueprop*)
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fun c_mkTrueprop A = Thm.capply cTrueprop A;
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(*Given an abstraction over n variables, replace the bound variables by free
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  ones. Return the body, along with the list of free variables.*)
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fun c_variant_abs_multi (ct0, vars) = 
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      let val (cv,ct) = Thm.dest_abs NONE ct0
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      in  c_variant_abs_multi (ct, cv::vars)  end
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      handle CTERM _ => (ct0, rev vars);
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(*Given the definition of a Skolem function, return a theorem to replace 
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  an existential formula by a use of that function.*)
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fun skolem_of_def def =  
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  let val (c,rhs) = Drule.dest_equals (cprop_of (Drule.freeze_all def))
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      val (ch, frees) = c_variant_abs_multi (rhs, [])
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      val (chil,cabs) = Thm.dest_comb ch
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      val {sign,t, ...} = rep_cterm chil
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      val (Const ("Hilbert_Choice.Eps", Type("fun",[_,T]))) = t
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      val cex = Thm.cterm_of sign (HOLogic.exists_const T)
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      val ex_tm = c_mkTrueprop (Thm.capply cex cabs)
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      and conc =  c_mkTrueprop (Drule.beta_conv cabs (Drule.list_comb(c,frees)));
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  in  prove_goalw_cterm [def] (Drule.mk_implies (ex_tm, conc))
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	    (fn [prem] => [ rtac (prem RS someI_ex) 1 ])
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  end;	 
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(*Converts an Isabelle theorem (intro, elim or simp format) into nnf.*)
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fun to_nnf thy th = 
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    th |> Thm.transfer thy
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       |> transform_elim |> Drule.freeze_all
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       |> ObjectLogic.atomize_thm |> make_nnf;
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(*The cache prevents repeated clausification of a theorem, 
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  and also repeated declaration of Skolem functions*)  (* FIXME better use Termtab!? *)
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val clause_cache = ref (Symtab.empty : (thm * thm list) Symtab.table)
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(*Declare Skolem functions for a theorem, supplied in nnf and with its name*)
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fun skolem thy (name,th) =
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  let val cname = (case name of "" => gensym "sko" | s => Sign.base_name s)
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      val (thy',axs) = declare_skofuns cname (#prop (rep_thm th)) thy
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  in (map skolem_of_def axs, thy') end;
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(*Populate the clause cache using the supplied theorems*)
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fun skolemlist [] thy = thy
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  | skolemlist ((name,th)::nths) thy = 
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      (case Symtab.lookup (!clause_cache) name of
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	  NONE => 
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	    let val (nnfth,ok) = (to_nnf thy th, true)  
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	                 handle THM _ => (asm_rl, false)
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            in
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                if ok then
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                    let val (skoths,thy') = skolem thy (name, nnfth)
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			val cls = Meson.make_cnf skoths nnfth
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		    in change clause_cache (Symtab.update (name, (th, cls)));
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			skolemlist nths thy'
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		    end
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		else skolemlist nths thy
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            end
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	| SOME _ => skolemlist nths thy) (*FIXME: check for duplicate names?*)
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(*Exported function to convert Isabelle theorems into axiom clauses*) 
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fun cnf_axiom (name,th) =
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    case name of
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	  "" => cnf_axiom_aux th (*no name, so can't cache*)
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	| s  => case Symtab.lookup (!clause_cache) s of
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	  	  NONE => 
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		    let val cls = cnf_axiom_aux th
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		    in change clause_cache (Symtab.update (s, (th, cls))); cls end
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	        | SOME(th',cls) =>
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		    if eq_thm(th,th') then cls
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		    else (*New theorem stored under the same name? Possible??*)
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		      let val cls = cnf_axiom_aux th
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		      in change clause_cache (Symtab.update (s, (th, cls))); cls end;
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fun pairname th = (Thm.name_of_thm th, th);
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fun meta_cnf_axiom th = 
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    map Meson.make_meta_clause (cnf_axiom (pairname th));
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(* changed: with one extra case added *)
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fun univ_vars_of_aux (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,body)) vars =    
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      univ_vars_of_aux body vars
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  | univ_vars_of_aux (Const ("Ex",_) $ Abs(_,_,body)) vars = 
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      univ_vars_of_aux body vars (* EX x. body *)
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  | univ_vars_of_aux (P $ Q) vars =
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      univ_vars_of_aux Q (univ_vars_of_aux P vars)
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  | univ_vars_of_aux (t as Var(_,_)) vars = 
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      if (t mem vars) then vars else (t::vars)
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  | univ_vars_of_aux _ vars = vars;
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fun univ_vars_of t = univ_vars_of_aux t [];
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fun get_new_skolem epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,tp,_)))  = 
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    let val all_vars = univ_vars_of t
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	val sk_term = ResSkolemFunction.gen_skolem all_vars tp
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    in
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	(sk_term,(t,sk_term)::epss)
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    end;
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(*FIXME: use a-list lookup!!*)
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fun sk_lookup [] t = NONE
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  | sk_lookup ((tm,sk_tm)::tms) t = if (t = tm) then SOME (sk_tm) else (sk_lookup tms t);
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(* get the proper skolem term to replace epsilon term *)
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fun get_skolem epss t = 
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    case (sk_lookup epss t) of NONE => get_new_skolem epss t
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		             | SOME sk => (sk,epss);
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fun rm_Eps_cls_aux epss (t as (Const ("Hilbert_Choice.Eps",_) $ Abs(_,_,_))) = 
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       get_skolem epss t
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  | rm_Eps_cls_aux epss (P $ Q) =
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       let val (P',epss') = rm_Eps_cls_aux epss P
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	   val (Q',epss'') = rm_Eps_cls_aux epss' Q
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       in (P' $ Q',epss'') end
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  | rm_Eps_cls_aux epss t = (t,epss);
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fun rm_Eps_cls epss th = rm_Eps_cls_aux epss (prop_of th);
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(* replace the epsilon terms in a formula by skolem terms. *)
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fun rm_Eps _ [] = []
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  | rm_Eps epss (th::thms) = 
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      let val (th',epss') = rm_Eps_cls epss th
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      in th' :: (rm_Eps epss' thms) end;
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(**** Extract and Clausify theorems from a theory's claset and simpset ****)
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(*Preserve the name of "th" after the transformation "f"*)
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fun preserve_name f th = Thm.name_thm (Thm.name_of_thm th, f th);
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(*Tags identify the major premise or conclusion, as hints to resolution provers.
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  However, they don't appear to help in recent tests, and they complicate the code.*)
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val tagI = thm "tagI";
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val tagD = thm "tagD";
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val tag_intro = preserve_name (fn th => th RS tagI);
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val tag_elim  = preserve_name (fn th => tagD RS th);
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fun rules_of_claset cs =
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  let val {safeIs,safeEs,hazIs,hazEs,...} = rep_cs cs
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      val intros = safeIs @ hazIs
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      val elims  = safeEs @ hazEs
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  in
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     debug ("rules_of_claset intros: " ^ Int.toString(length intros) ^ 
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            " elims: " ^ Int.toString(length elims));
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     if tagging_enabled 
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     then map pairname (map tag_intro intros @ map tag_elim elims)
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     else map pairname (intros @ elims)
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  end;
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fun rules_of_simpset ss =
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  let val ({rules,...}, _) = rep_ss ss
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      val simps = Net.entries rules
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  in 
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      debug ("rules_of_simpset: " ^ Int.toString(length simps));
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      map (fn r => (#name r, #thm r)) simps
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  end;
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fun claset_rules_of_thy thy = rules_of_claset (claset_of thy);
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fun simpset_rules_of_thy thy = rules_of_simpset (simpset_of thy);
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fun claset_rules_of_ctxt ctxt = rules_of_claset (local_claset_of ctxt);
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fun simpset_rules_of_ctxt ctxt = rules_of_simpset (local_simpset_of ctxt);
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(**** Translate a set of classical/simplifier rules into CNF (still as type "thm")  ****)
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(* classical rules *)
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fun cnf_rules [] err_list = ([],err_list)
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  | cnf_rules ((name,th) :: thms) err_list = 
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      let val (ts,es) = cnf_rules thms err_list
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      in  (cnf_axiom (name,th) :: ts,es) handle  _ => (ts, (th::es))  end;  
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(**** Convert all theorems of a claset/simpset into clauses (ResClause.clause) ****)
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fun addclause (c,th) l =
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  if ResClause.isTaut c then l else (c,th) :: l;
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(* outputs a list of (clause,thm) pairs *)
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   405
fun clausify_axiom_pairs (thm_name,thm) =
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    let val isa_clauses = cnf_axiom (thm_name,thm) 
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   407
        val isa_clauses' = rm_Eps [] isa_clauses
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   408
        val clauses_n = length isa_clauses
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	fun make_axiom_clauses _ [] []= []
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	  | make_axiom_clauses i (cls::clss) (cls'::clss') =
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	      addclause (ResClause.make_axiom_clause cls (thm_name,i), cls') 
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	                (make_axiom_clauses (i+1) clss clss')
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    in
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   414
	make_axiom_clauses 0 isa_clauses' isa_clauses		
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   415
    end
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   416
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   417
fun clausify_rules_pairs [] err_list = ([],err_list)
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  | clausify_rules_pairs ((name,thm)::thms) err_list =
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      let val (ts,es) = clausify_rules_pairs thms err_list
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   420
      in
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   421
	  ((clausify_axiom_pairs (name,thm))::ts, es) 
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	  handle THM (msg,_,_) =>  
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		  (debug ("Cannot clausify " ^ name ^ ": " ^ msg); 
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		   (ts, (thm::es)))
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   425
             |  ResClause.CLAUSE (msg,t) => 
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   426
                  (debug ("Cannot clausify " ^ name ^ ": " ^ msg ^
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   427
                          ": " ^ TermLib.string_of_term t); 
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   428
		   (ts, (thm::es)))
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   430
      end;
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dfe264950511 Moved some of the clausify functions from ATP/res_clasimpset.ML to res_axioms.ML.
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   431
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   433
(*Setup function: takes a theory and installs ALL simprules and claset rules 
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   434
  into the clause cache*)
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   435
fun clause_cache_setup thy =
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   436
  let val simps = simpset_rules_of_thy thy
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   437
      and clas  = claset_rules_of_thy thy
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   438
  in skolemlist clas (skolemlist simps thy) end;
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   439
  
16563
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   440
val clause_setup = [clause_cache_setup];  
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   441
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   442
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(*** meson proof methods ***)
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   444
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   445
fun cnf_rules_of_ths ths = List.concat (#1 (cnf_rules (map pairname ths) []));
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   446
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fun meson_meth ths ctxt =
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  Method.SIMPLE_METHOD' HEADGOAL
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    (CHANGED_PROP o Meson.meson_claset_tac (cnf_rules_of_ths ths) (local_claset_of ctxt));
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val meson_method_setup =
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 [Method.add_methods
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  [("meson", Method.thms_ctxt_args meson_meth, 
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    "The MESON resolution proof procedure")]];
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end;